Rapid Influence Maximization on Social Networks: The Positive Influence Dominating Set Problem

Abstract

Motivated by applications arising on social networks, we study a generalization of the celebrated dominating set problem called the Positive Influence Dominating Set (PIDS). Given a graph G with a set V of nodes and a set E of edges, each node i in V has a weight bi, and a threshold requirement gi. We seek a minimum weight subset T of V, so that every node i not in T is adjacent to at least gi members of T. When gi is one for all nodes, we obtain the weighted dominating set problem. First, we propose a strong and compact extended formulation for the PIDS problem. We then project the extended formulation onto the space of the natural node-selection variables to obtain an equivalent formulation with an exponential number of valid inequalities. Restricting our attention to trees, we show that the extended formulation is the strongest possible formulation, and its projection (onto the space of the node variables) gives a complete description of the PIDS polytope on trees. We derive the necessary and sufficient facet-dening conditions for the valid inequalities in the projection and discuss their polynomial time separation. We embed this (exponential size) formulation in a branch-and-cut framework and conduct computational experiments using real-world graph instances, with up to approximately 2.5 million nodes and 8 million edges. On a test-bed of 100 real-world graph instances, our approach finds solutions that are on average 0.2% from optimality and solves 51 out of the 100 instances to optimality. Summary of Contribution: In influence maximization problems, a decision maker wants to target individuals strategically to cause a cascade at a minimum cost over a social network. These problems have attracted significant attention as their applications can be found in many different domains including epidemiology, healthcare, marketing, and politics. However, computationally solving large-scale influence maximization problems to near optimality remains a substantial challenge for the computing community, which thus represent significant opportunities for the development of operations-research based models, algorithms, and analysis in this interface. This paper studies the positive influence dominating set (PIDS) problem, an influence maximization problem on social networks that generalizes the celebrated dominating set problem. It focuses on developing exact methods for solving large instances to near optimality. In other words, the approach results in strong bounds, which then provide meaningful comparative benchmarks for heuristic approaches. The paper first shows that straightforward generalizations of well-known formulations for the dominating set problem do not yield strong (i.e., computationally viable) formulations for the PIDS problem. It then strengthens these formulations by proposing a compact extended formulation and derives its projection onto the space on the natural node-selection variables, resulting in two equivalent (stronger) formulations for the PIDS problem. The projected formulation on the natural node-variables contains a new class of valid inequalities that are shown to be facet-defining for the PIDS problem. These theoretical results are complemented by in-depth computational experiments using a branch-and-cut framework, on a testbed of 100 real-world graph instances, with up to approximately 2.5 million nodes and 8 million edges. They demonstrate the effectiveness of the proposed formulation in solving large scale problems finding solutions that are on average 0.2% from optimality and solving 51 of the 100 instances to optimality.